ďđĘĤ:
N¬¢¢'XM÷j>İ­tı83S
PDFĖĉīĪĬěø83÷HZ÷ġĬĐÞĀĴ
http://www.ecei.tohoku.ac.jp/hariyama/id-2/1NöïÚñ
n 83÷ĢĬĤġĬĐ
ÄÏÏÌ°¨¨ÑÑÑ§ÂÀÂÅ§ÏËÄËÇÐ§¾À§ÆÌ¨Ä¾ÍÅÒ¾É¾¨÷HZ÷ġĬĐÞĀĠĩēīĘĬďĦī÷N£¹³´97¤ąĖĉīĪĬěòàúçĳòàĂì
âQúòöćėĠĪĬě-ĳHZöGíĆòäÚĳĝĔčīĲÅÌ¾ÁõôòGíĆòÚëìÚñýÿÚòçĳLqV÷ĊėĒīđąEåñÚĂ÷òĲ
LqVÿāø/ß2õÚòçĳĘđĚøLqVą oçãóĳ2

'XM÷j>£­t¤n  ÷ďđĘĤ×¢ĎğďđĘĤ÷|ûĄèn ďđĘĤ÷M×ĎğďđĘĤ÷M÷|ûĄèxõM÷j>⇒÷ďđĘĤ÷j>÷[jö1^yĲ2^yß(`/,öĲs÷1^/2^÷)
3

'XMn a z
n ;Ĳrz£ª^y÷Wýxõ9¤n ª^ăĲª^ûzn «^ăn Uăz4ôĆõj>ąùĂ÷ĵn !cMj>ØĜĈċđĚ}$ØģĬě}$n đĘėĠ=v
Ø U &ò÷đĘėĠö0çĂ

1t=0

•=>
t=0ò÷÷síßāöôăìâzRá=çĂÞt •-6 5t>>0÷-6k@òĲn]öôăìâðÚñÚĂÞ5'XM÷úóüM1^û2^ăUăİĤĖUıª£ª¥Î»¤ª¥Î»ÖÊ«º«¥«ÓÖÊÎ¥ÖÊ«Â¦¸Î

1/T1/T;r1^ă·Î

·ÎKģĬě}$ 1/K
÷ČĈī20dB/dec -20dB/dec-20dB/dec
π1/T00ģĬě}$ 2-π-π÷p022ImImImReĜĈċđĚ
1ReRe}$
đĘėĠ=v

20dB/dec

0-40dB/dec
00ωco=ωn1/T-π/20-πIm0ImIm11163%0t0t0 Tt0t0tt0 L
6

a z
 X(s)K Y(s)KĴ_÷.Msą úõÚĭĭ,#÷ Ĵ DCRMøĲ
I(s)V(s)G(s) =I/V=1/R7a z÷ģĬě}$dBČĈīø20log|G(jω)|= 20logK20logK

gdB[dB]!cMMøG(jω)= KωpøMAßõÚ÷ò0φ[rad]

π/2
0.1K

K 10K

ω

-π/28;z X(s)sdx (t )y (t ) =dt

 Y(s)Y(s)=sX(s)(P.51÷ħĠħđ)Kÿā)

9;z÷!cM=vĴĜĈċđĚ}$!cMMøĲG(jω)=jωjω→∞0 ω=010;z÷!cM=vĴģĬě}$!cMMøĲG(jω)=jωČĈīgdBø
gdB=20 log(ω)
pφøφ=π/2[rad]

0.1

10.1

110

ω[rad]-20π/2

φ[rad]

ČĈīø|G(jω)|=|jω|=ω

gdB[dB]20àĴ
20dB/dec-π/210ω[rad]
11;z÷F8£LqVöõÚ¤;zöazß|ûĄäăë( X(s)Ks

 Y(s)MøG(s)=Ks12

£;¥a¤z÷!cM=vĴģĬě}$!cMMøĲG(jω)=jKω

dBČĈīgdBøgdB=20 log(Kω)=20 logω+20logKKω=1÷óàgdB=0pφøφ=π/2[rad]0.1/K 1/K
10/K

ω[rad]-20
π/2

φ[rad]

ČĈīø|G(jKω)|=|jKω|=Kω

gdB[dB]20

àĴ20dB/dec-π/20.1/K 1/K10/K

ω[rad]
13h~Ĵµ£Î¤±ª©Î÷ģĬě}$÷[9ąJâgdB[dB]

20

ω
-20

φ[rad]

π/2

ω-π/2
14

rz
 X(s)

1/sty(t ) = ò x(t )dt0

 Y(s)Y(s)=(1/s)X(s)(ħĠħđ)Kÿā)16

rz÷!cM=vĴĜĈċđĚ}$!cMMøĲ
G(jω)=(1/(jω))=(1/ω)(-j)

j0 ω→∞

ω=017

rz÷!cM=vĴģĬě}$1
j=−!cMMø G( jω ) =jωωpφøÕ±¢¦Ô¨«¼Í¾Á½gdB[dB]ČĈīgdBøgdB=20 log(1/ω)=20(log1-logω)= -20log(ω)

200.1

10.1

1

10

ω[rad]-20
π/2

φ[rad]

ČĈīø|G(jω)|=1/ωàĴ-20dB/dec

-π/2

10

ω[rad]
18

rz÷F8£LqVöõÚ¤rzöa

zß|ûĄäăë( X(s)K1/s

 Y(s)MøK
G(s)=s19£a¥r¤z÷!cM=vĴģĬě}$!cMMø G( jω ) = K = − j Kjωpφøφ= ¦Ô¨«[rad]gdB[dB]dBČĈīgdBøgdB=«©¢ÈËÃ£·¨Ö¤±«©ÈËÃ·¢¦ «©ÈËÃÖ(K/ω=1÷óàgdB=0)-20dB/dec

20K10K

ω

0.1K
-20

π/2

φ[rad]

ČĈīø|G(jω)|=K/ωω-π/20.1K

K 10K

ω
20µ£Î¤±ª©¨Î÷ģĬě}$÷[9ąJâgdB[dB]

20

ω
-20

φ[rad]π/2ω-π/221h~£¹§¯ª¤G(s) X(s)a50/s

K Y(s)zóĲrzÞĀõĂďđĘĤ

ω=500[rad/sec]öÝÚñĲďđĘĤ ÷ČĈīą1öçĂëü÷ĲK÷ ąbüÿ23v
 X(s)50/sK Y(s)ďđĘĤ÷Mø50
50 K
G( s) = ´ K =ss
ω=500÷óà÷ČĈīß1ÿā
50 K
50 K
| G ( jv ) |v =500 =||v =500 ==1500v
ÿîñĲK=1024

ª^ăz11+sT

 X(s) Y(s) T:U-M1^ăz÷Vi) RC#RCVoRC#öÝâĂĲ
(f%Vi)ó(čīęīĎ÷%Vo)÷M

T=RC25ª^ăz÷!cM=vĴĜĈċđĚ}$!cMMø1
1 - jvTG( jv ) ==1 + jvT 1 + (vT ) 2.xĲyøéăêă
1-vT
y=
x=
2
2
1 + (vT )1 + (vT )ÿîñ2 2v T1
22 2v T = -1 y =2
1 + (vT )
xj
ω=01

ω=∞

ω+21ö
æ1ö
æ
2
çx- ÷ + y =ç ÷2ø
è2ø
è
26

2

ª^ăz÷ČĈīgÁ²
ČĈīø
1
1
1| G (v ) |===
1 + jvT | 1 + jvT |1 + v 2T 2íõûöČĈīßW+ ÷(1/√2)öõĂω: O!cMωcoČĈīgdBø20 log | G (v ) |= 20 log

11 + v 2T 2

12 22 2= 20(log(1) - log(1 + v T )) = -10 log(1 + v T )227ª^ăz÷pÕpφø

- vT-1ÐG(v ) = tan () = tan (-vT )1-1j
10φ-ωT28ª^ăz÷ģĬě}$°ČĈīgÁ²!cMMø G( jv ) =ČĈīgdBø1
1 - jvT=
1 + jvT 1 + (vT ) 220 log | G(v ) |= -10 log(1 + v T )22nωT<<1Ĵ 1+ω2T2=1ÿā2ååÚòçß[9ąJâĂÿÛöåñÝág dB = -10 log1 = 0nωT=1g dB = -10 log(1 + 1) = -10 log 2nωT>> 1Ĵ 1+ω2T2=ω2T2ÿā
g dB = -10 log(v 2T 2 ) = -20 log(vT )29

gdB[dB]ª^ăz÷ģĬě}$°ČĈīgÁ²1/T

10/T

ω

-20-20[dB/dec]30

ª^ăz÷ģĬě}$ĴpÕÐG(v ) = tan (-vT )-1nωT<<1Ĵ ωT→0ÿā
f =0nωT=1
f = tan (-1) = -1

p4nωT>> 1Ĵ ωT→∞ÿā
pf =231

φ[rad]ª^ăz÷ģĬě}$°pÕ1/T

10/T

ω

-π/4-π/232µ£Î¤±ª¨£ª¥ª©Î¤÷ģĬě}$÷[9ąJâgdB[dB]

20

ω
-20

φ[rad]π/2

ω-π/233ª^ăz÷đĘėĠ=vħĠħđ&ò÷đĘėĠ=vøĲ

P.51(2)(đĘėĠ÷ħĠħđ)K)ąlÚĂãóöÿā

1
1 1
1
1Y ( s) =X ( s) ==1 + sT
1 + sT s s( s + 1 ) TTħĠħđ)KP.51(8)ÿāα=0,β=1/Tóåñt
T

1 1−0ty(t) =(e − e ) = 1 − e
T 1 −0T−−tT35đĘėĠ=v
1t=TĴU=vß
W{ ÷63Į
(1-e-1≒0.63)

0.63

T:U-M1^ăy÷Uj>ąçĝħĥĬĕ
0 TU t36mõU=vÒ£Ï¤÷büP£¤ X(s)G(s) Y(s)ďđĘĤ÷M G(s)X(s)ö0åñĲs &öÝâĂY(s)ąbüĂ

Y (s) = G(s)X(s)X(s)=1/s (đĘėĠ=v)
X(s)=1 (ĈīĝĨđ=v)

Y(s)ąħĠħđ)KçĂ
−1y(t) = L [Y (s)]38ª^ăzRVi1CVo
0.63

RC#öÝâĂĲ
(f%Vi)ó
(čīęīĎ%Vo)÷M

T=RC0 TčīęīĎ÷tó
39

¡­ĳ«£¹§¯®¤
120
G( s) =s + 120
ã÷1^ăyöïÚñ÷"ÚövÜÿ
(a)O!cMωcoąbüÿ
(b)ωcoöÝâĂpφcoąbüÿ(c)đĘėĠąÜë(ĲßW{÷63ĮöõĂUąbüÿëìåĲO!cMóøĲßĲW+!cMòÙĂ

÷1/√2öõĂ
40

v£¾¤!cMMø120
G( jv ) =jw + 120
W+120
120
| G( jv ) |==jv + 120v 2 + 1202

øω=0÷óà1òÙĂĳO!cMòøĲČĈīßW+÷1/√2óõĂ÷ò

120

1=ÿāĲωco=120[rad/sec]
222v + 120
41v£¿¤
O!cMòø

120
1- jG( jv co ) ==j120 + 1202

j1/20

φ-j/2

ÿîñĲΦ= -π/442v£À¤1^ăz÷]g9ÿā
120
1
G( s) ==s + 120 1 + s 1120
ÿîñĲU-MT=1/120
ß63ĮöõĂ÷øĲt=T=1/120[sec]÷óà

43ª^ûz X(s)1+sT

 Y(s)1^ăz÷`ó,ą1^ûz÷

R
¶Å

L Vo

RL#öÝâĂĲ
(d¶Å)ó(čĈĨ÷%Vo)÷M

T=L/R
44

ª^ûz÷!cM=vĴĜĈċđĚ}$!cMMøG( jv ) = 1 + jvTjω+.xĲyøéăêă

x =1y = vT1ω=045

ª^ûz÷ČĈīgÁ²!cMMøG( jv ) = 1 + jvTČĈīø| G(v ) |=| 1 + jvT |= 1 + (vT )22ååÚòçß
[9ąJâĂÿÛöåñÝá

ČĈīgdBø20 log | G(v ) |= 20 log 1 + v 2T 2 = 10 log(1 + v 2T 2 )1^ăzóuß46

ª^ûz÷pÕpφø1) = tan (vT )-11^ăzóuß

ωTφvTÐG(v ) = tan (-1j0147

ª^ûz÷ģĬě}$°ČĈīgÁ²20[dB/dec]

gdB[dB]20

ω
1/T

10/T48ª^ûz÷ģĬě}$°pÕπ/2

φ[rad]

π/4

ω
1/T

10/T49µ£Î¤±ª¥ª©Î÷ģĬě}$÷[9ąJâgdB[dB]20

ω-20φ[rad]π/2ω-π/250đĘėĠ=v
ħĠħđ&ò÷đĘėĠ=vøĲP.51(2)ÿā
11Y ( s) = (1 + sT ) X ( s) = (1 + sT ) = T +ssħĠħđ)KP.51(1)Ĳ(2)ÿā

y(t) = T σ (t) +1
ì¥(t = 0)ããòĲs (t ) = íî 0(t ¹ 0)53đĘėĠ=vy(t) = Tσ (t) +1∞

10t54

h~^÷M÷ģĬě}$(ČĈīj>)÷[9ąÞâK (1+ sT1 )G(s) =s (1+ sT2 )ëìåĲT1>>T2>>(1/K)
ĞīĚĴ'XMöįåĄè

56

v'XMöįåĄèK (1 + sT1 ) K 1
G(s) ==(1 + sT1 )s (1 + sT2 ) s 1 + sT2
g1g2

g3

g1: r+ag2: Uăzg3: ^ûz
g1,g2,g3÷ČĈīj>÷[9÷w→G(s)÷ČĈīj>÷[9(wąçĂ<øõÚ!!)57

v
g dB = 20 log G ( s)
æ 1 öæKö
÷÷ + 20 log(1 + sT1 )= 20 logç ÷ + 20 logççèsøè 1 + sT2 ø
T1>>T2>>(1/K)ÿāĲ(1/T1)<(1/T2)<K58

v
ω<1/T1⇒g1÷û(g2,g3=0)ĳ

K
1 1
T1 T2

ω 1/T1ω<1/T2⇒g1óg3òBíeåÙÛ
(g2=0)ĳ)iå1/T2ω⇒g2öÿā59

h~ĴČĈīj>ÞĀMąI-çĂW1pzą-İ^÷đħĈěòTı-20dB/dec
20

0.1

1 10ω

ĞīĚĴ4ÞĀj-åñÚàúåþÛ
61

W1pz
2 ï÷ďđĘĤ÷ČĈīj>ßæ≠ãăĀ÷ďđĘĤ÷
pj>ßæW1py: ČĈīj>ßæ\ØõM÷õÞòpăßWý1äÚý÷W1py⇒ ČĈīj>ÞĀpj>ß
?ö-úĂ
ďđĘĤąYçĂöøW1pyą-çĂãóß*Ú

64